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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>To satisfy the initial condition, set <span class="process-math">\(t=0\text{,}\)</span> we get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,0)=\sum_{n=1}^{\infty}C_n \sin\frac{n\pi x}{L}=f(x),\qquad 0\leq x\leq L.
\end{equation*}
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<p class="continuation">In other words, we need to choose the coefficients <span class="process-math">\(C_n\)</span> so that the series of sine functions converges to the initial temperature distribution <span class="process-math">\(f(x)\)</span> for <span class="process-math">\(0 \leq x \leq L\text{.}\)</span> The series is just the Fourier sine series for <span class="process-math">\(f\)</span> and its coefficients are given by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_n=\frac{2}{L}\int_{0}^L f(x)\sin\frac{n\pi x}{L}\textrm{d}x,\qquad n=1,2,3,\cdots
\end{equation*}
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